Synthesis of Digital PID Controllers for Discrete-Time Systems With Guaranteed Non-Overshooting Transient Response

In this paper, we present a new method of synthesizing digital PID controllers for discrete-time, Linear Time Invariant (LTI) Systems satisfying a class of transient response specifications. The problem of synthesizing a controller to achieve desirable transient specifications, such as requiring the transient response to be within an allowable range of overshoot, can be carried out as a problem of guaranteeing the impulse response of an appropriate closed loop error transfer function to be non-negative. An earlier result by the authors provides necessary and sufficient conditions for the impulse response of a discrete-time transfer function to be non-negative in terms of the requirement of a sequence of polynomials to be sign-invariant on the interval [1, ∞). An application of this result to the error transfer function yields a sequence of polynomials which are required to be sign-invariant on [1, ∞) but whose coefficients are polynomial functions of the controller gains k₁, k₂ and k₃.